smooth algebraic groups over $\mathbb{R}$ or $\mathbb{C}$ :: Lie algebras

smooth algebraic groups over R (any commutative ring) :: Formal group law

in which I tried to answer this question, and ended up with “the Lazard ring doesn’t quite work,” which makes sense in retrospect, as the Lazard ring is not associated with any particular formal group law. When I say “formal group law” I mean “1-d formal group law.”

What is a formal group law? It’s an expression of the group structure of G in an infinitesimal neighborhood of the origin. At the Midwest Topology Seminar, talking with Paul V. and Dylan Wilson, I have a somewhat more satisfying answer.

What is a Lie algebra? It’s an expression of the group structure of $G$ at the FIRST infinitesimal neighborhood of the origin. In characteristic 0, this extends to a definition of the group structure in an infinitesimal neighborhood of the origin by the Baker Campell Hausdorff formula.

Specifically, what is the universal enveloping algebra of a formal group law? It’s the formal group law itself! Well, more specifically:

Universal enveloping algebra :: Lie algebra

Functions on Formal group law :: Formal group law

According to wikipedia, “The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring S used above is just a completion of this Hopf algebra.”

A formal group law already has a Hopf algebra structure. This is just the cogroup on formal power series $R[[x]]$ induced by the formal group law, $f$, that is,

$$R[[x]] \to R[[x]] \widehat{\otimes} R[[x]]$$

$$x \mapsto f(1 \otimes x, x \otimes 1)$$

This is already complete! We’re a formal group law so we’re already completed at the origin! And, if we are a 1-d formal group law, we’re always commutative (unless our ring is nilpotent), so this is promising.

This post assumes that you are familiar with the definition of Lie group/algebra, and that you are comfortable with the Lazard ring. Note: This is less of an expository post and more of an unfinished question.

Why care about formal group laws? Well, we want to study smooth algebraic groups, but Lie algebras fail us in characteristic p (for example, $\frac{d}{dx}(x^p) = 0$), so, rather than a tangent bundle, we take something closer to a jet bundle.

Lie algebras are to smooth groups over $\mathbb{R}$ or $\mathbb{C}$ as formal group laws are to smooth algebraic groups over any ring $R$.

I want to apply this analogy! I want this deeply. I’m trying to puzzle out how to see if this analogy is deep or superficial. How deep does the rabbit hole go? Let’s look at an example.

Given the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, we might think of this as a deformation of the Symmetric algebra:

$$U_{\epsilon}(\mathfrak{g}) := T(\mathfrak{g})/(x \otimes y – y \otimes x – \epsilon[x, y])$$

$$\mathbb{C}[\mathfrak{g}^*] \simeq Symm[\mathfrak{g}] := T(\mathfrak{g})(x \otimes y – y \otimes x)$$

Action on all of this is the adjoint action, that is, the action which takes an element $g$ of a Lie group $G$ sends $X \mapsto gXg^{-1}$. Orbits of this action stratify the dual Lie algebra, and there is a symplectic form that lives on each orbit.

This quick post assumes basic knowledge of Lie algebras and category equivalence. I am new to category theory, and appreciative of constructive feedback.

We commonly study smooth manifolds, e.g. Lie groups, by studying their tangent spaces. Since the product map induces a map from one tangent space to another, we can oftentimes just consider the tangent space to a Lie group at the identity. This tangent space can be equipped with a structure induced by the group structure, called a Lie algebra structure.

The Theorems of Lie

The assignment Lie : $G \to $Lie$(G)$ is functorial. The theorems of Lie in their modern incarnation emerge out of the attempt to see how close this functor is to being an equivalence of categories. Note that we are working in Diff.

Lie proved that the category of local real Lie groups is equivalent to the category of finite-dimensional real Lie algebras.

This equivalence was extended to global cases by Cartan: the category of real Lie algebras is equivalent to the category of simply-connected Lie Groups.

Note: We cannot drop the condition of being simply connected for $G$, as, for example, $G = S^1$ and $G = \mathbb{R}$ have the same Lie algebras, but are not isomorphic.

Lie I: Groups $\to$ Algebras

The assignment $G \mapsto$ Lie$(G)$ induces a functor Lie: LieGrp $\to$ LieAlg and for each morphism $g:G \to H$ of Lie groups the following diagram commutes:

The Lie algebra homomorphism Lie$(g)$ is equivalent to the first order infinitesimal of the group homomorphism.

Lie II: Do You Even Lift, Lie?
Alegbras $\to$ Groups

Let $G$ and $H$ be Lie groups with Lie algebras Lie$(G)$ and Lie$(H)$, with a Lie algebra homomorphism $f:$Lie$(G) \to$Lie$(H)$.

For notational convenience, we denote Lie$(G)$ as the lowercase gothic letter $\mathfrak{g}$.

Lie II states that there exists a unique morphism $F$ lifting $f$

such that $f=$Lie$(F)$.

Lie-Cartan III: Groups $\leftrightarrow$ Algebras

The functor Lie cannot be inverted because locally isomorphic Lie groups have isomorphic Lie algebras. However, we can invert Lie on the subcategory of simply connected Lie groups. The essential surjectivity of this functor is the third theorem.

For every finite-dimensional real Lie algebra $\mathfrak{g}$ there exists a Lie group $G$ with Lie algebra $\mathfrak{g}$.

Group(oid) theory is the study of symmetry. When we are dealing with objects that appear symmetric, group theory assists with analysis of these objects. The label of “symmetric” is applied to anything which stays invariant under some transformations.

This can apply to geometric figures (the unit circle, $S^1$, is highly symmetric, for it is invariant under any rotation): This also applies to more abstract objects such as functions: the trigonometric functions $sin(\theta)$ and $cos(\theta)$ are invariant when we replace $\theta$ with $\theta+\tau$.

Both functions are periodic with period $\tau$. Periodicity is a type of symmetry. Important points for $cos(\theta)$ and $sin(\theta)$ in terms of their period $\tau$:

The subject of Fourier analysis is concerned with representing a wave-like ($\tau$-periodic) function as a combination of simple sine waves (simpler $\tau$-periodic functions). More formally, it decomposes any periodic function into the sum of a set of oscillating functions (sines and cosines, or equivalently, complex exponentials).

Fourier analysis is central to spectroscopy, passive sonar, image processing, x-ray crystallography, and more. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.

Fourier analysis is a fusion of analysis, linear algebra, and group theory. [Source]

Furthermore, group theory is the beating heart of physics.

We care a great deal about group representations, especially of Lie groups, since these representations often point the way to the “possible” physical theories. Examples of the use of groups in physics include the Standard Model and gauge theory.

Modern particle physics would not exist without group theory; in fact, group theory predicted the existence of many elementary particles before they were found experimentally. [Source]

Studying symmetry allows us to discover the laws of the universe.

What is a group?

More formally,

Homomorphisms Between Groups

A group homomorphism is a structure preserving map $\psi$ from a group $(G, \cdot)$ to a group $(H, *)$, satisfying:

which means $\psi(e_G) = e_H$

which means $\forall g_1, g_2 \in G$, $\psi(g_1 \cdot g_2) = \psi(g_1) * \psi(g_2)$

An exponential $exp: (\mathbb{R}, +) \rightarrow (\mathbb{R}^+, *)$ is a morphism from the reals under addition to the positive reals under multiplication.

The operation on the domain $\mathbb{R}$ is addition, while the operation on the range $\mathbb{R}$ is mulitplication.

Thus, to show $exp$ is a homomorphism, I must show that $$\forall x, y \in \mathbb{R}, exp(x+y) = exp(x)*exp(y)$$ Recall that $exp(x+y) \equiv e^{x+y}$, and $exp(x) * exp(y) = e^xe^y$, so the equation to be verified comes down to the familiar identity $e^{x+y} = e^xe^y$, thus $exp$ is a homomorphism!

Note that a group together with group homomorphisms form a category.

Interested?

Group Explorer (free) allows you to autogenerate visualizations of groups, homomorphisms, subgroup lattices, and more.

Visual Group Theory: Nathan Carter’s expository text is a beautifully illustrated, gentle introduction to groups, ending in quintics.

A group in which the objects are matrices and the group operation is matrix multiplication is called a linear group (or matrix group).

Since in a group every element must be invertible, the most general linear groups are the groups of all invertible matrices of a given size, called the general linear groups $GL(n)$.

Any property of matrices that is preserved under matrix multiplication and inverses can be used to define further linear groups.

Elements of $GL(n)$ with determinant 1 form a subgroup called the special linear group $SL(n)$. Orthogonal matrices ($M^{T}M = I$) form the orthogonal group $O(n)$. The elements of the special orthogonal group $SO(n)$ are both orthogonal and have determinant 1.

Linear groups pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry.

Geometry is the study of invariants of the action of a matrix group on a space.

Particle physics, 4 dimensional topology, and Yang-Mills connections are inter-related theories based heavily on matrix groups, particularly on a certain double cover between two matrix groups (which I’ll cover in Clifford’s Road to Spinors).

Quantum computing is based on the group of unitary matrices. “A quantum computation, according to one widely used model, is nothing but a sequence of simple unitary matrices. One starts with a small repitoire of some 2×2 and some 4×4, and combines them to generate, with arbitrarily high precision, an approximation to any desired unitary transformation on a huge vector space.” – William Wootters

Riemannian geometry relies heavily on matrix groups, in part because the isometry group of any compact Riemmanian manifold is a matrix group.

Circle $\cong$ SO(2)

We will begin with the simplest example of a smooth group: the group of proper rotations in 2 dimensions, isomorphic to SO(2).

The compositions of the rotations by two angles $\theta_1$ and $\theta_2$ corresponds to a rotation of the angle $\theta_1 + \theta_2$.

The map $\cdot(\theta_1, \theta_2) = \theta_1 + \theta_2$, takes two elements of the group as arguments and returns another element of the group ($\cdot: G \times G \to G$) is smooth (continuous and differentiable).

We have another important property: the proper rotations are periodic. Rotations by angles differing by multiples of $\tau$ are periodic.

$$R(\theta + \tau) = R(\theta)$$

As a manifold, this group is the circle $S^1$.

The continuity and differentiability of the product map has a very profound consequence: the elements of the group are determined by the elements close to the identity (the infinitesimal transformations).

Indeed, if we wish to determine how a rotation $R(\theta)$ depends on $\theta$, we look at how $R$ changes with respect to infinitesimal change of $\theta$.

For groups of linear transformation on a space, we can use the language of differential operators or that of matrix components, according to our taste and convenience.

This rotation group $SO(2)$ can be regarded as a group of transformations acting on its group manifold $S^1$.

Smooth groups are conventionally referred to as Lie groups, after Sophus Lie.

Why Study Smooth Groups?

Groups elegantly represent the symmetries of geometric objects. For example, the finitely many symmetries of polygons are captured by the Dihedral groups.

The infinitely many symmetries of circles require more sophistication. Observe that an axis of symmetry exists for every angle in $[0,\tau]$, so there should exist a continuous map from $[0,\tau]$ into any group representing the symmetries of a circle. The pristine algebraic nature of a group fails to capture this notion of continuity, so we much enrich it to obtain the smooth groups.

Motivated by geometry, smooth groups merge the perspectives of algebra and analysis, tying together these normally disparate fields with great efficacy.

Not convinced by their beauty and simplicity alone?

Smooth groups are useful:

The study of irreducible representations of the smooth group $SO(3)$ lead to an explanation of the Periodic Table.

The study of irreducible representations of the smooth group $SU(2)$ naturally leads to the Dirac equation describing the electron.

The classification of the unitary representations of the Poincare group earned Wigner the 1963 Nobel prize in physics.

The Standard Model, which unifies 3 of the the 4 fundamental forces in nature, is described by the smooth group $SU(3)\times SU(2) \times U(1)$.

What is a Smooth Group?

A group object in the category of smooth manifolds is called a smooth group.

Equivalently, a group $G$ is a smooth group if it is a manifold, and the product and inverse operations $\cdot:G\times G\rightarrow G$ and $^{-1}:G \rightarrow G$ are smooth maps.

Representation: A Special Kind of Homomorphism

SO(3) describes the rotational symmetries of 3-dimensional Euclidean space. SO(3) acts on $\mathbb{R}^3$ (any element of SO(3) defines a linear transformation of $\mathbb{R}^2$.

We can generalize this to say a group $G$ acts on a vector space $V$ if there is a map $\phi$ from $G$ to linear transformations of $V$ s.t. $\forall v \in V; g,h \in G$

$$\phi(gh)v = \phi(g)\phi(h)v$$

The map $\phi$ is called a representation of $G$ on $V$; it’s really just a special kind of homomorphism.

Recall that the general linear group $GL(V)$ is the group of all invertible linear transformations of $V$. A representation of our smooth group $G$ on $V$ is merely a homomorphism

$$\phi: G \rightarrow GL(V)$$

We can think of a representation as a linear action of a group/algebra on a linear space (since to every $g \in G$ there is an associated linear operator $\phi(g)$ which acts on a linear space $V$).

Recall that a smooth group is a group object in the category of Diff. When $G$ is a smooth group, we usually restrict our attention to representations of $G$ in $GL(V)$, where $V$ is finite dimensional and $\phi$ is a smooth map. Since we are operating on a smooth manifold, we can apply the tools of differential geometry.

Symmetries of Differential Equations

The group of symmetries of a differential equation $(x,y,…, u, )$ is the set of all transformations of the independent variables $(x,y, …)$ and of the dependent variables $(u, v, …)$ that transform solutions to solutions.

Lie proved that if the group of symmetries is solvable then the differential equation can be integrated by quadratures, and he found a method to compute the symmetries. For more on this, the keyword is “heat equation.”

A Note for the Adventurous

If we’re feeling algebraic, we can consider the set of invertible matrices over an arbitrary unital ring $R$. Thus $GL_n: R \rightarrow GL_n(R)$ becomes a presheaf of groups on $Aff = Ring^{op}$.

Postscript: Adding the unitary group to the visualization, while establishing completeness, disrupts the symmetric aesthetic.